*3.4. Pressure Wave Propagation*

The speed of sound in gases is dependent on the temperature. Under the assumption of isentropic flow, the speed of sound is estimated by *c* = γ*RT*. γ and *R* are the ratios of specific heats and the individual gas constant with the assigned values of 1.4 and 287.058 J/kg.K, respectively; at an air temperature of 300 K, *c* = 347.1 m/s. Under atmosphere pressure and linear wave assumption, the pressure waves propagate at the speed of sound [31]. However, in the Hyperloop system, the reference pressure is reduced to 1/1000 atm and the pressure waves have high amplitude, causing the nonlinear wave phenomenon. Therefore, the pressure wave propagation speeds are greater than the speed of sound. Figure 12 compares the pressure wave propagation speeds induced at the front and rear end of the pod with the speed of sound. A higher BR creates higher pressure wave propagation speed at the front end of the pod *vs*, *f ront*, but it barely affects the pressure wave propagation speed at the end of the pod (*vs*, *rear*).

(**b**) Pod-relative coordinate

**Figure 12.** Forward and backward pressure wave propagation speeds observed in (**a**) absolute coordinates (the reference frame is fixed) and (**b**) pod-relative coordinates (the reference frame moves with the pod). The red dashed line represents the speed of sound (347.1 m/s).

Figure 12a shows the values of pressure wave propagation speed (*vs*) in absolute coordinates. The range of *vs*, *f ront* is from 355 to 473 m/s, which is much higher than the normal speed of sound of 347.1 m/s. As *vP* increases, *vs*, *f ront* increases steeply. In the rear end of the pod, *vs*, *rear* did not change much and maintained a speed around the speed of sound. At *vP* of 350 m/s, the differences between speed of sound and *vs* for forward and backward flows are 26.6% and 0.6%, respectively. Note that in this study, the analysis of *vs* is conducted under ideal operating conditions, where the tube walls are smooth and straight and without the application of a vacuum pump.

As shown in Figure 12b, the pod-relative coordinates and the absolute coordinates are vastly different. *vs*,*rear* is much higher than *vs*, *f ront*. This is because the directions of *vP* and *vs*, *f ront* are the same, in contrast to the direction of *vs*, *rear*, which is the opposite of *vP*. That means as *vP* increases, the difference between *vP* and the compression wave speed decreases, whereas the difference between *vP* and the expansion wave speed increases.

Figure 13 illustrates the pressure variation along the tube axis and the pod surface. As *vP* increases, the pressure magnitude at the front of the pod significantly increases, whereas the pressure behind the pod slightly decreases. Compression waves are generated in front of the pod and propagate mainly in the forward direction faster than the speed of sound. These compression waves cause

the pressure variation in front of the pod to fluctuate abruptly and generate a high-pressure region. Meanwhile, expansion waves are generated behind the pod, and propagate both in the forward and backward directions of the pod at around the speed of sound. The forward expansion waves pass the pod surface and merge with the incident compression waves, decreasing the pressure of the front waves. This phenomenon is probably observed at lower speeds, namely, 100 m/s and 150 m/s. This pressure wave distribution is similar to the variation in the high-speed train–tunnel system presented by Zonglin et al. [13]. When *vP* increases, the compressibility of air and the compression wave propagation speed increase, and consequently alter the forms of the high-pressure region in front of the pod. Figure A4 (Appendix C) describes this high-pressure region in more detail. By contrast, the position of the low-pressure region shows only a minor change. The oblique shockwaves caused by the reflection of waves between the tube walls and the pod tail are illustrated in Figure 13 for *vP* ≥ 250 m/s.

**Figure 13.** Pressure wave distribution along the centerline of the tube and pod surface at *t* = 1.0 s. Four cases of selected pod speeds under *Ptube* = 101.325 Pa, *L* = 43 m, and BR = 0.36 are presented. Pod moves from right to left.

Figure 14 shows the pressure variation along the tube axis in the front and rear of the pod (depicted in Figure 15) for *t* = 0.2–1.0 s. Compression waves are generated ahead of the pod and propagate in the same direction, whereas expansion waves propagate in the opposite direction. The following equation presents the pressure ratio for flows across a normal shock [27,31]:

$$\frac{p\_2}{p\_1} = \frac{2\gamma \mathcal{M}\_s^2 - (\gamma - 1)}{\gamma + 1} \tag{19}$$

with

$$M\_s = \frac{\upsilon\_s}{\sqrt{\gamma RT\_1}}\tag{20}$$

**Figure 14.** Pressure distributions of line 1 (the axis line in front of the pod) and line 2 (the axis line at the rear of the pod) for *t* = 0.2–1.0 (*vP* = 300 m/s, BR = 0.36, *Ptube* = 101.325 Pa, *Lpod* = 43 m). Pod moves from right to left. Red triangles represent the tube pressure of 101.325 Pa.

**Figure 15.** Descriptions of line 1 and line 2 from Figure 12.

The term *p*2 in Equation (19) is the pressure term of the front shockwave shown in Figure 16a. Figure 16a compares the pressure magnitude of the front shockwave between BRs of 0.25 and 0.36. The results indicate that a higher BR produces a higher front shockwave pressure. In Figures 12–14, stronger compression waves form in front of the pod with increase of *vP*. This behavior of front waves is similar to the phenomenon in the normal shockwave theory. Hence, Figure 16b compares the compression wave propagation speed generated in front of the pod from the simulation with the ones calculated by Equations (19) and (20). There is a good agreemen<sup>t</sup> between the two results. Therefore, the compression wave generated in front of the pod in this study conforms to the normal shockwave theory.

(**b**) Speed of front shockwave

**Figure 16.** Pressure magnitude and speed of front shockwave. The compression wave is well-matched with the normal shockwave theory. (**a**) Pressure magnitude of compression wave traveling in the forward direction of the pod. (**b**) Comparison of compression wave propagation speeds calculated using the simulation and normal shockwave equations.
